Question

lim n–> infinity prove that (√(n+1)-√n)=0​

Answer 1

Answer:

$\displaystyle\lim_{n\rightarrow\infty}\bigl(\sqrt{n+1}-\sqrt{n}\bigr)\\\\=\lim_{n\rightarrow\infty}\frac{\bigl(\sqrt{n+1}-\sqrt{n}\bigr)\bigl(\sqrt{n+1}+\sqrt{n}\bigr)}{\sqrt{n+1}+\sqrt{n}}\\\\=\lim_{n\rightarrow\infty}\frac{(n+1)-n}{\sqrt{n+1}+\sqrt{n}}\\\\=\lim_{n\rightarrow\infty}\frac1{\sqrt{n+1}+\sqrt{n}}\\\\=0$

where the final limit is equal to zero because the denominator increases without bound as n tends to infinity, while the numerator is constant.